On a surprising covariance identity

Abstract

This paper deals with the fact that the covariance of the integer and fractional parts of a sum of n random numbers chosen independently and randomly on (0,1) is- 1/12, for all n >2. While the result is of interest, it is the three different ways proposed to arrive at the result that are particularly instructive. These proofs highlight the usefulness of: (i) different results in elementary probability, and (ii) computational techniques, which are also of interest on their own.